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Theorem pm14.123c 27542
Description: Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.123c  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  /\  E. z E. w ph )
) )
Distinct variable groups:    w, A, z    w, B, z
Allowed substitution hints:    ph( z, w)    V( z, w)    W( z, w)

Proof of Theorem pm14.123c
StepHypRef Expression
1 pm14.123a 27540 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  /\  [. A  /  z ]. [. B  /  w ]. ph )
) )
2 pm14.123b 27541 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
)  /\  [. A  / 
z ]. [. B  /  w ]. ph )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  /\  E. z E. w ph )
) )
31, 2bitrd 245 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B ) )  /\  E. z E. w ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   [.wsbc 3153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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